### 机器学习代写|主成分分析作业代写PCA代考| Basis for Subspace Tracking

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 机器学习代写|主成分分析作业代写PCA代考|Extension or Generalization of PCA

It can be found that the above-mentioned algorithms only focused on eigenvector extraction or eigen-subspace tracking with noncoupled rules. However, a serious speed stability problem exists in the most noncoupled rules [28]. This problem is that in noncoupled PCA rules the eigen motion in all directions mainly depends on the principal eigenvalue of the covariance matrix; thus, numerical stability and fast convergence can only be achieved by guessing this eigenvalue in advance [28]; in noncoupled MCA rules the speed of convergence does not only depend on the minor eigenvalue, but also depend on all other eigenvalues of the covariance matrix, and if these extend over a large interval, no suitable learning rate may be found for a numerical solution that can still guarantee stability and ensure a sufficient speed of convergence in all eigen directions. Therefore, the problem is even more severe for MCA rules. To solve this common problem, Moller proposed some coupled PCA algorithms and some coupled MCA algorithms based on a special information criteria [28]. In coupled rules, the eigen pair (eigenvector and eigenvalue) is simultaneously estimated in coupled equations, and the speed of convergence only depends on the eigenvalue of its Jacobian. Thus, the dependence of the eigenvalues on the covariance matrix can be eliminated [28]. Recently, some modified coupled rules have been proposed [48].

It is well known that the generalized eigen decomposition (GED) plays very important roles in various signal processing applications, e.g., data compression, feature extraction, denoising, antenna array processing, and classification. Though PCA, which is the special case of GED problem, has been widely studied, the adaptive algorithms for the GED problem are scarce. Fortunately, a few efficient online adaptive algorithms for the GED problem that can be applied in real-time applications have been proposed [49-54]. In [49], Chaterjee et al. present new adaptive algorithms to extract the generalized eigenvectors from two sequences of random vectors or matrices. Most algorithms in literatures including [49] are gradient-based algorithms $[50,51]$. The main problem of this type of algorithms is slow convergence and the difficulty in selecting an appropriate step size which is essential: A too small value will lead to slow convergence and a too large value will lead to overshooting and instability. Rao et al. [51] have developed a fast recursive least squares (RLS)-like, not true RLS, sequential algorithm for GED. In [54], by reinterpreting the GED problem as an unconstrained minimization problem via constructing a novel cost function and applying projection approximation method and RLS technology to the cost function, RLS-based parallel adaptive algorithms for generalized eigen decomposition was proposed. In [55], a power method-based algorithm for tracking generalized eigenvectors was developed when stochastic signals having unknown correlation matrices are observed. Attallah proposed a new adaptive algorithm for the generalized symmetric eigenvalue problem, which can extract the principal and minor generalized eigenvectors, as well as their corresponding subspaces, at a low computational cost [56]. Recently, a fast and
numerically stable adaptive algorithm for the generalized Hermitian eigenvalue problem (GHEP) was proposed and analyzed in [48].

Other extensions of PCA also include dual-purpose algorithm [57-64], the details of which can be found in Chap. 5 , and adaptive or neural networks-based SVD singular vector tracking $[6,65-70]$, the details of which can be found in Chap. $9 .$

## 机器学习代写|主成分分析作业代写PCA代考|Concept of Subspace

Definition 1 If $\boldsymbol{S}=\left{\boldsymbol{u}{1}, \boldsymbol{u}{2}, \ldots, \boldsymbol{u}{m}\right}$ is the vector subset of vector space $\boldsymbol{V}$, then the set $\boldsymbol{W}$ of all linear combinations of $\boldsymbol{u}{1}, \boldsymbol{u}{2}, \ldots, \boldsymbol{u}{m}$ is called the subspace spanned by $\boldsymbol{u}{1}, \boldsymbol{u}{2}, \ldots, \boldsymbol{u}{\boldsymbol{m}}$, namely $$\boldsymbol{W}=\operatorname{Span}\left{\boldsymbol{u}{1}, \boldsymbol{u}{2}, \ldots, \boldsymbol{u}{m}\right}=\left{\boldsymbol{u}: \boldsymbol{u}=\alpha_{1} \boldsymbol{u}{1}+\alpha{2} \boldsymbol{u}{2}+\cdots+\alpha{m} \boldsymbol{u}{m}\right}$$ where each vector in $\boldsymbol{W}$ is called the generator of $\boldsymbol{W}$, and the set $\left{\boldsymbol{u}{1}, \boldsymbol{u}{2}, \ldots, \boldsymbol{u}{m}\right}$ which is composed of all the generators is called the spanning set of the subspace. A vector subspace which only comprises zero vector is called a trivial subspace. If the vector set $\left{\boldsymbol{u}{1}, \boldsymbol{u}{2}, \ldots, \boldsymbol{u}_{m}\right}$ is linearly irrespective, then it is called a group basis of $W$.

Definition 2 The number of vectors in any group basis of subspace $W$ is called the dimension of $W$, which is denoted by $\operatorname{dim}(W)$. If any group basis of $W$ is not composed of finite linearly irrespective vectors, then $W$ is called an infinite-dimensional vector subspace.

Definition 3 Assume that $\boldsymbol{A}=\left[a_{1}, a_{2}, \ldots, a_{n}\right] \in \boldsymbol{C}^{\mathrm{m} \times n}$ is a complex matrix and all the linear combinations of its column vectors constitute a subspace, which is called column space of matrix $A$ and is denoted by $\operatorname{Col}(\boldsymbol{A})$, namely
$$\operatorname{Col}(A)=\operatorname{Span}\left{a_{1}, a_{2}, \ldots, a_{n}\right}=\left{\boldsymbol{y} \in \boldsymbol{C}^{m}: \boldsymbol{y}=\sum_{j=1}^{n} \alpha_{j} a_{j}: \alpha_{j} \in \boldsymbol{C}\right}$$
Row space of matrix $A$ can be defined similarly.

## 机器学习代写|主成分分析作业代写PCA代考|Subspace Tracking Method

The iterative computation of an extreme (maximal or minimum) eigen pair (eigenvalue and eigenvector) can date back to 1966 [72]. In 1980, Thompson proposed a LMS-type adaptive algorithm for estimating eigenvector, which correspond to the smallest eigenvalue of sample covariance matrix, and provided the adaptive tracking algorithm of the angle/frequency combing with Pisarenko’s harmonic estimator [14]. Sarkar et al. [73] used the conjugate gradient algorithm to track the variation of the extreme eigenvector which corresponds to the smallest eigenvalue of the covariance matrix of the slowly changing signal and proved its much faster convergence than Thompson’s LMS-type algorithm. These methods were only used to track single extreme value and eigenvector with limited application, but later they were extended for the eigen-subspace tracking and updating methods. In 1990, Comon and Golub [6] proposed the Lanczos method for tracking the extreme singular value and singular vector, which is a common method designed originally for determining some big and sparse symmetrical eigen problem $A x=\lambda x[74]$.

The earliest eigenvalue and eigenvector updating method was proposed by Golub in 1973 [75]. Later, Golub’s updating idea was extended by Bunch et al. [76, 77 , the basic idea of which is to update the eigenvalue decomposition of the covariance matrix after every rank-one modification, and then go to the matrix’s latent root using the interlacing theorem, and then update the place of the latent root using the iterative resolving root method. Thus, the eigenvector can be updated. Later, Schereiber [78] introduced a transform to change a majority of complex number arithmetic operation into real-number operation and made use of Karasalo’s subspace mean method [79] to further reduce the operation quantity. DeGroat and

Roberts [80] developed a numerically stabilized rank-one eigen structure updating method based on mutual Gram-Schmidt orthogonalization. Yu [81] extended the rank-one eigen structure update to block update and proposed recursive update of the eigenvalue decomposition of a covariance matrix.

The earliest adaptive signal subspace tracking method was proposed by Owsley [7] in 1978. Using the stochastic gradient method, Yang and Kaveh [18] proposed a LMS-type subspace tracking algorithm and extended Owsley’s method and Thompson’s method. This LMS-type algorithm has a high parallel structure and low computational complexity. Karhumen [17] extended Owsley’s idea by developing a stochastic approaching method based on computing subspace. Like Yang and Kaveh’s extension of Thompson’s idea to develop an LMS-type subspace tracking algorithm, Fu and Dowling [45] extended Sarkar’s idea to develop a subspace tracking algorithm based on conjugate gradient. During the recent 20 years, eigen-subspace tracking and update has been an active research field. Since eigen-subspace tracking is mainly applied to real signal processing, these methods should be fast algorithms.

## 机器学习代写|主成分分析作业代写PCA代考|Extension or Generalization of PCA

PCA 的其他扩展还包括双用途算法 [57-64]，其详细信息可在第 1 章中找到。5、自适应或基于神经网络的SVD奇异向量跟踪[6,65−70]，其详细信息可以在第 1 章中找到。9.

## 机器学习代写|主成分分析作业代写PCA代考|Concept of Subspace

\operatorname{Col}(A)=\operatorname{Span}\left{a_{1}, a_{2}, \ldots, a_{n}\right}=\left{\boldsymbol{y} \in \boldsymbol {C}^{m}: \boldsymbol{y}=\sum_{j=1}^{n} \alpha_{j} a_{j}: \alpha_{j} \in \boldsymbol{C}\right}\operatorname{Col}(A)=\operatorname{Span}\left{a_{1}, a_{2}, \ldots, a_{n}\right}=\left{\boldsymbol{y} \in \boldsymbol {C}^{m}: \boldsymbol{y}=\sum_{j=1}^{n} \alpha_{j} a_{j}: \alpha_{j} \in \boldsymbol{C}\right}

## 机器学习代写|主成分分析作业代写PCA代考|Subspace Tracking Method

Roberts [80] 开发了一种基于互 Gram-Schmidt 正交化的数值稳定秩一特征结构更新方法。Yu [81]将秩一特征结构更新扩展到块更新，并提出了协方差矩阵的特征值分解的递归更新。

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